We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula as a combination of two ingredients - an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelöf hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Half integral weight forms
- Special values of L-functions
- Waldspurger correspondence